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Weighted Sobolev Approximation Rates for Neural Networks on Unbounded Domains
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In this work, we consider the approximation capabilities of shallow neural networks in weighted Sobolev spaces for functions in the spectral Barron space. The existing literature already covers several cases, in which the spectral Barron space can be approximated well, i.e., without curse of dimensionality, by shallow networks and several different classes of activation function. The limitations of the existing results are mostly on the error measures that were considered, in which the results are restricted to Sobolev spaces over a bounded domain. We will here treat two cases that extend upon the existing results. Namely, we treat the case with bounded domain and Muckenhoupt weights and the case, where the domain is allowed to be unbounded and the weights are required to decay. We first present embedding results for the more general weighted Fourier-Lebesgue spaces in the weighted Sobolev spaces and then we establish asymptotic approximation rates for shallow neural networks that come without curse of dimensionality.
Forward citations
Cited by 4 Pith papers
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On Generalized Barron Spaces for Shallow Neural Networks
A phi-weighted generalized Barron space B^phi_sigma is defined for shallow networks and shown to embed into Sobolev spaces, to unify known Barron-type spaces, and to yield approximation and regularization bounds.
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Beyond Universal Approximation Theorems: Algorithmic Uniform Approximation by Neural Networks Trained with Noisy Data
An explicit randomized training pipeline is claimed to yield uniform approximators from noisy data with minimax-optimal trainable parameters, but key sample-complexity claims are algebraically reversed and the proof s...
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Approximation Rates for Metaplectic Neural Networks
Metaplectic Barron spaces admit N^{-1/2} Sobolev approximation rates using chirped ridge neurons, and a metaplectic PINN beats a plain PINN on harmonic-oscillator Schrödinger benchmarks.
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Approximation Rates in Fr\'echet Metrics: Barron Spaces, Paley-Wiener Spaces, and Fourier Multipliers
Two theorems give sufficient shallow-network width to reach a prescribed error in a Fréchet metric of semi-norms, applied to exponential spectral Barron, Gelfand-Shilov, and bandlimited (Paley-Wiener type) symbol classes.
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